Aryabhata: the Aryabhatiya · Chapter 3 of 8

#Numbers, squares and shapes

The second section of the Aryabhatiya, the Ganita ("calculation"), is a compact textbook of mathematics in 33 verses. Its first ten verses cover numbers, roots, areas and volumes.

#The opening (Ganita 1)

Having paid reverence to Brahman, the Earth, the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms, Aryabhata sets forth here the science which is honored at Kusumapura (Ganita 1).

Notice that the Earth is listed among the heavenly bodies, beside the planets.

#The names of the decimal places (Ganita 2)

The numbers eka (one), dasha (ten), shata (hundred), sahasra (thousand), ayuta (ten thousand), niyuta (hundred thousand), prayuta (million), koti (ten million), arbuda (hundred million) and vrinda (thousand million) are from place to place each ten times the preceding (Ganita 2).

This is a statement of the decimal place system: each place is worth ten times the one before. The Indian system of ten digits with place value, which later travelled through the Arab world to Europe, is one of India's greatest gifts to the world; this verse is one of its early clear statements.

#Squares and cubes (Ganita 3)

A square (varga) is a square figure, its area, or the product of two equal numbers. A cube (ghana) is the product of three equal numbers, or a solid with twelve edges (Ganita 3).

#Square roots (Ganita 4)

One should always divide the non-square place by twice the square root of the preceding square place. After subtracting the square of the quotient from the square place, the quotient will be the square root to the next place (Ganita 4, Clark, simplified).

This is the method of finding square roots digit by digit, still taught in some schools. The number is split into pairs of digits from the right ("square places" and "non-square places"), and the root is found one digit at a time.

Example: the square root of 54,756.

1. Split into pairs from the right: 5 | 47 | 56.
2. The largest square not more than 5 is 4 (2²). First digit of the root: 2. Remainder: 5 − 4 = 1.
3. Bring down the next digit: 14. Divide by twice the root so far (2 × 2 = 4): 14 ÷ 4 = 3 (rounded down). Second digit: 3. Remainder 14 − 12 = 2. Bring down the next digit: 27. Subtract the square of 3: 27 − 9 = 18.
4. Bring down the next digit: 185. Divide by twice the root so far (2 × 23 = 46): 185 ÷ 46 = 4. Third digit: 4. Remainder 185 − 184 = 1. Bring down the last digit: 16. Subtract the square of 4: 16 − 16 = 0.

The square root is 234 (and 234 × 234 = 54,756).

#Cube roots (Ganita 5)

Aryabhata gives the similar digit-by-digit method for cube roots, splitting the number into groups of three digits and using three times the square of the root found so far as the divisor (Ganita 5). For example, the cube root of 12,167 is 23.

#Areas and volumes (Ganita 6–9)

Triangle. "The area of a triangle is the product of the perpendicular and half the base" (Ganita 6). Correct: area = ½ × base × height.

Pyramid. "Half the product of this area and the height is the volume of a solid which has six edges" (Ganita 6) — that is, a triangular pyramid. This is wrong: the correct volume is one-third, not one-half, of the base area times the height.

Circle. "Half of the circumference multiplied by half the diameter is the area of a circle" (Ganita 7). Correct: area = ½ C × r = πr².

Sphere. "This area multiplied by its own square root is the exact volume of a sphere" (Ganita 7). This is wrong. For a sphere of radius 1, the rule gives π × √π ≈ 5.57, while the true volume is 4π/3 ≈ 4.19 — the rule is about a third too large. Aryabhata calls it "exact"; it is not. (Later Indian mathematicians, for example Bhaskara II, gave a correct rule.)

Trapezium. The perpendiculars from the point where the two diagonals cross to the two parallel sides are found by multiplying each side by the height and dividing by the sum of the sides (Ganita 8).

Any figure. "The area of any plane figure is found by determining two sides and then multiplying them together" (Ganita 9) — that is, by reducing it to rectangles.

These two errors are worth remembering. Even the greatest scientists make mistakes, and the honest course is to say so. Aryabhata's achievement does not depend on his being right about everything.

#π (Ganita 10)

Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000 (Ganita 10).

(100 + 4) × 8 + 62,000 = 62,832. Divided by 20,000: 3.1416.

This is correct to four decimal places (π = 3.14159...). And Aryabhata uses the word asanna, "approximately" (or "nearly"). Some historians have read this as a sign that he knew the value could not be expressed exactly — that π is irrational — though that reading is debated. What is certain is that he gave one of the best values of π known anywhere at the time.

#Summary

• Ganita 2 states the decimal place system: each place ten times the previous.
• Aryabhata gives digit-by-digit methods for square roots (Ganita 4) and cube roots (Ganita 5).
• His triangle and circle areas are correct (Ganita 6, 7); his pyramid volume (half instead of a third) and sphere volume (about a third too large) are wrong.
• His value of π is 62,832 ÷ 20,000 = 3.1416, called "approximate" (Ganita 10).

#Exercises

1. Using Ganita 4, find the square root of 1,764.
2. What is the area of a triangle with base 10 and height 7, by Ganita 6?
3. By Ganita 6, what volume would a pyramid with base area 30 and height 10 have? What is the correct volume?
4. Work out Aryabhata's value of π from Ganita 10.
5. Why might the word "approximately" in Ganita 10 be important?

Answers: 1. 42 (17 | 64: first digit 4, remainder 1; 16 ÷ 8 = 2, remainder 0; 04 − 4 = 0). 2. 35. 3. By Aryabhata's rule ½ × 30 × 10 = 150; correctly ⅓ × 30 × 10 = 100. 4. (100 + 4) × 8 + 62,000 = 62,832; ÷ 20,000 = 3.1416. 5. Because it shows Aryabhata knew his value was not exact, and some historians read it as an awareness that π cannot be written exactly as a ratio.